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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 81
Chapter 4, Problem 81

Find the inverse of f(x)=(x−10)/(x+10).

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Start with the function equation: \(y = \frac{x - 10}{x + 10}\).
To find the inverse, swap \(x\) and \(y\): \(x = \frac{y - 10}{y + 10}\).
Multiply both sides by \((y + 10)\) to eliminate the denominator: \(x(y + 10) = y - 10\).
Distribute \(x\): \(xy + 10x = y - 10\).
Group all terms involving \(y\) on one side and factor \(y\) out: \(xy - y = -10 - 10x\), then \(y(x - 1) = -10(1 + x)\), and finally solve for \(y\): \(y = \frac{-10(1 + x)}{x - 1}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. If f(x) maps x to y, then its inverse f⁻¹(x) maps y back to x. Finding an inverse involves solving the equation y = f(x) for x in terms of y.
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Graphing Logarithmic Functions

Solving Rational Equations

Rational equations involve ratios of polynomials. To find the inverse of a rational function, you often need to solve for the variable by clearing denominators and isolating terms. This requires careful algebraic manipulation to avoid extraneous solutions.
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Introduction to Rational Equations

Domain and Range Considerations

When finding inverses, it's important to consider the domain and range of the original function, as the inverse's domain and range are swapped. For rational functions, restrictions like division by zero must be accounted for to ensure the inverse is valid.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

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Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x2+x−4)/(2x2−5x)

Textbook Question

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

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Textbook Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x

Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0

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Textbook Question

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0